Oblique Shock Calculator - Turn Angle and Downstream Mach

An oblique shock calculator turns a measured upstream Mach number and shock wave angle into the post-shock flow properties across a compression corner. An oblique shock is a compression shock inclined at an angle to the upstream flow, formed when supersonic flow meets a turning surface such as a wedge or compression ramp. It turns the flow by the deflection angle theta and raises static pressure, density, and temperature while reducing stagnation pressure through a small entropy jump.

• • Aerodynamics coursework: Solve textbook and homework problems for compressible flow and gas dynamics without redoing the theta-beta-M algebra.
• • Inlet and diffuser sizing: Estimate the pressure rise and downstream Mach from an inlet ramp on a supersonic engine.
• • Wind-tunnel and CFD checks: Compare measured shock stand-off and turn angles against a quick solver before running CFD.
• • Concept and trade studies: Run parametric sweeps over wave angle and Mach to size wedges, ramps, and supersonic nozzles.

The shock is an abrupt jump in flow properties, thin enough to treat as a discontinuity line where mass, momentum, and energy are conserved but entropy rises.

Because the shock is inclined, only the velocity component normal to the shock sees the discontinuity, which is why the solver reports Mx and My alongside M1 and M2.

An attached oblique shock is the canonical pressure-drag source on a supersonic wedge, so the same wedge area plus the post-shock static pressure feeds directly into the Drag Equation Calculator for the wave-drag contribution.

The calculator takes three inputs and applies the oblique shock equations in sequence. Each output uses the upstream conditions together with the wave angle beta, the deflection angle theta, and the specific heat ratio gamma.

• M₁: Upstream Mach number ahead of the shock, supplied by the user.
• β (beta): Shock wave angle measured from the upstream flow to the shock line, supplied by the user.
• γ (gamma): Specific heat ratio cp/cv of the gas, 1.4 for air at room temperature.
• θ (theta): Deflection angle the flow turns through as it crosses the shock.
• Mₓ: Normal-component upstream Mach M₁ sin(β), the value used for the normal shock relations.
• Mᵧ: Normal-component downstream Mach M₂ sin(β - θ).

Once theta is known, the downstream Mach M2 follows from M2n divided by sin(beta - theta), where M2n is the normal-shock downstream Mach evaluated at Mx.

The pressure, density, and temperature ratios use the standard normal shock relations evaluated at Mx, and the stagnation pressure ratio uses the Rayleigh-Pitot product at Mx and gamma.

According to NASA Glenn Research Center, Oblique Shocks, an oblique shock is a shock wave inclined to the upstream flow, the deflection angle comes from resolving the flow parallel and perpendicular to the shock, and for any (M1, theta) pair two roots exist where the supersonic "weak shock" branch is the one that forms in nature while the subsonic "strong shock" branch only appears under special conditions.

Stagnation pressure is conserved along a streamline in isentropic flow, so the same idea behind the Bernoulli Equation Calculator explains why the oblique shock stagnation pressure ratio drops in the loss term.

Four ideas do almost all the work in this oblique shock calculator. Once you can name them and point to where each appears in the equations, every shock problem becomes the same template.

These four ideas frame the engineering decisions: weak vs strong picks the right branch, the detachment condition tells you when an attached shock can no longer hold, and the normal-component view links every oblique shock to the simpler normal shock relations.

Mass, momentum, and energy are conserved across the shock plane, so the same conservation law behind the Conservation of Momentum Calculator pins down the post-shock normal Mach and the deflection angle once M1, beta, and gamma are known.

The default example loads at Mach 5 with a 20 degree wave angle in air, the same setup as the worked example.

1. 1 Enter the upstream Mach number M₁: Type the Mach number ahead of the shock; must be greater than 1, usually between 1.2 and 10.
2. 2 Enter the shock wave angle β: Angle between the shock line and the upstream flow in degrees, larger than arcsin(1/M₁) and smaller than 90.
3. 3 Set the specific heat ratio γ: Use 1.4 for air, 1.3 for CO2, 1.66 for helium, or 1.667 for monatomic gases like argon.
4. 4 Read the deflection angle θ: First row of the results, equal to the wedge or ramp angle for an attached shock.
5. 5 Inspect the Mach numbers and ratios: Remaining rows list Mx, M2, My, p₂/p₁, ρ₂/ρ₁, T₂/T₁, and p₀₂/p₀₁ for sizing.
6. 6 Reset for the next case: Press Reset to restore the Mach 5, beta 20 default, then change one input at a time to study sensitivity.

An aerospace student sizes a 15 degree compression ramp on the inlet of a supersonic air-breathing engine. With M1 = 4, beta = 27 degrees, gamma = 1.4, the student reads theta about 15 degrees, downstream Mach about 2.93, and pressure ratio about 3.68.

When you need the actual static pressures upstream and downstream of the shock, plug the ratio from this calculator together with p1 from the Gas Laws Calculator so the ideal gas law converts the ratios into engineering units.

The calculator collapses the oblique shock table, the normal shock relations, and the theta-beta-M chart into one result panel.

• • Skip the iterative solve: Returns theta from the theta-beta-M relation in one step instead of graphing or iterating the curve.
• • Stay consistent across ratios: Pressure, density, temperature, and stagnation pressure ratios come from the same Mx so the readouts agree.
• • Switch gases without re-deriving: Change gamma between 1.3 and 1.66 to move between air, combustion products, and noble gases.
• • Quick CFD and textbook cross-check: Compare a CFD case, a wind-tunnel schlieren, or a homework answer against a fast solver.
• • Stay inside the valid shock range: The solver flags any wave angle at or below the Mach angle mu = arcsin(1/M1), so you catch an invalid input before reading a wrong deflection angle.
• • Set downstream conditions fast: Pair the density ratio with an upstream density from a separate tool to get the absolute downstream density for sizing.

The calculator is most useful as a first-pass tool for sizing supersonic inlets, ramps, and wedges and as a teaching aid for theta-beta-M homework. For high-fidelity design, pair it with a CFD case or the textbook chart.

To convert the density ratio into an absolute downstream density, multiply the upstream density from the Air Density Calculator by rho2 over rho1 to size ducts, inlets, and engine faces downstream of the shock.

The oblique shock relations are exact for a steady, planar, adiabatic shock in a perfect gas with constant gamma. Real flows move a few percent away from that ideal case.

• • The relations assume a perfect gas with constant cp, cv, and gamma; high-temperature air or dissociated flows need a variable gamma model.
• • The theta-beta-M equation is single-valued for any (M1, beta) pair above the Mach angle, so this calculator takes beta as input and returns theta directly; an inverse solver that takes theta would need to pick between weak and strong roots.

For high-Mach or viscous flows, treat the readouts as a baseline and apply a CFD or real-gas correction.

According to Anderson, Modern Compressible Flow (4th edition), the theta-beta-M relation tan(theta) = 2 cot(beta) (M1^2 sin^2(beta) - 1) / (M1^2 (gamma + cos 2 beta) + 2) ties upstream Mach, shock wave angle, and deflection angle for a perfect gas and is the equation every oblique shock solver, chart, and simulator must satisfy.

According to NACA Report 1135, the oblique shock equations and theta-beta-M chart used to size supersonic inlets, ramps, and wedges were assembled in NACA-1135 in 1951 and remain the canonical reference for the tabulated pressure, density, temperature, and Mach ratios across an oblique shock.

Once the static pressure jump is known, plug the downstream density and viscosity into the Reynolds Number Calculator to check whether the adverse pressure gradient across the shock stays inside the boundary layer or separates it.